( n + 1 ) -Dimensional reduced differential transform method for solving partial differential equations

In this paper, a variable-coefficient Korteweg-de Vries (vcKdV) equation which was proposed to describe the propagation of weakly nonlinear waves in water of variable depth is investigated in detail. First- ly, the Lie point symmetries and some group invariant solutions of this equation are presented. Furthermore, the integrability property of this variable-coefficient equation is studied following the Painlev e test of partial differential equations and an integrability condition is obtained. Following this condition, some meaningful exact solutions are obtained via the truncated auto-Bcklund transformation and the mapping transformation method. e test

http://internonlinearscience.org/upload/papers/IJNS%20Vol%2014%20No%203%20Paper%209%20Symmetry,%20Integrability%20and%20Exact%20Solutions%20of%20a%20Variable-Coefficient%20Korteweg-de%20Vries%20(vcKdV)%20Equation.pdf

Symmetry, Integrability and Exact Solutions of a Variable-Coefficient Korteweg-de Vries (vcKdV) Equation

This paper presents a novel method for new exact solutions of the nonlinear evolution equations. The new method is based on the property that many traveling wave solutions of nonlinear evolution equations can be written as a polynomial of two elementary functions which satisfy a projective Riccati equation, and theories of Gr¨ obner bases. Some examples are presented to illustrate the correctness and efficiency of our method.

http://www.internonlinearscience.org/upload/papers/20111228033058951.pdf

A Computational Algebraic Approach to New Exact Solutions of the Nonlinear Evolution Equations

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A Computational Algebraic Approach to New Exact Solutions of the Nonlinear Evolution Equations